Lemma B.1 (Monotonicity of Symmetric Norms, see e.g. Proposition IV.1.1 in [1]}
) 

If \(\Vert  \cdot \Vert _{\mathrm {sym}}\)  is a symmetric norm and \(x,y\in \mathbb {R}^d\) 
satisfy that for all \(i \in [d]\) 
, \(|x_i|\le |y_i|\) , then \(\Vert  x \Vert _{\mathrm {sym}} \le \Vert  y \Vert _{\mathrm {sym}}\) .

